
We see at once that \(a_{\frac{1}{2}} = 0\) , because the only supermultiplet that could yield \(h=\frac{1}{2}\)  states is the hypermultiplet \(\hat{{\cal B}}_{\frac{1}{2}}\) , violating assumption (i).

\(a_1\)  counts the number of \(\hat{{\cal B}}_{1}\)  multiplets, which are the only non-free multiplets with \(h=1\) . This basic fact is used throughout the paper.

\(a_{\frac{3}{2}}\)  gets a positive contribution from the number of \(\hat{{\cal B}}_{\frac{3}{2}}\)  multiplets and a negative contribution from the number of additional supercurrent multiplets (if any). One often has an a priori opinion about the number of supersymmetries of the theory, and thus can read off the number of Higgs operators with \(R = \frac{3}{2}\) . (They are necessarily generators of the chiral ring since they cannot be composites of operators with lower \(R\) ).

\(a_2\)  gets a variety of contributions. There are first of all the affine Kac-Moody algebra states \(\lbrace  J^a_{-2} |0 \rangle , J_{-1}^a J_{-1}^b | 0 \rangle \rbrace \) ,
where \(a, b\)  are adjoint indices of the flavor symmetry algebra. We denote their number
by \(a_2^{\rm KM}\) .To determine \(a_2^{\rm KM}\) , one must of course take care of subtracting possible null states, which are determined if one knows the levels of the (simple factors of the) affine current algebra. In the absence of null states, \(a_2^{\rm KM} = a_1 + a_1 (a_1+1)/2\) . If the central charge satisfies the Sugawara condition (REF ), the 4d stress tensor multiplet \(\hat{{\cal C}}_{0 (0 ,0)}\)  is already accounted for by the 2d Sugawara construction [1]}, [2]}. If the Sugawara condition is not obeyed, there must be an independent generator with \(h=2 \)  to account for the 4d stress tensor. Finally, there can be additional generators of type
\(\hat{\cal B}_2\)  (bosonic), \({\cal D}_{1(0,0)}\) , \(\overline{{\cal D}}_{1(0,0)}\)  (fermionic), or
\({\cal D}_{\frac{1}{2}(0,\frac{1}{2})}\) , \(\overline{{\cal D}}_{\frac{1}{2}(\frac{1}{2},0)}\)  (bosonic). It is important to note that
CPT invariance of the 4d theory implies that \({\cal D}_{R(0,j)}\)  and \(\overline{{\cal D}}_{R(j,0)}\)  multiplets must come in pairs.
All in all then, we haveHere \(\# \hat{\cal B}_2\)  denote the number of additional \(\hat{\cal B}_2\)  multiplets, on top of the ones that correspond to the composites \(J_{-1}^a J_{-1}^b | 0 \rangle \) . These additional \(\hat{\cal B}_2\)  operators are generators of the Higgs chiral ring.
\(a_2 = a_2^{\rm KM} + n^{\rm T} + \# \hat{\cal B}_2 - 2\, \# {\cal D}_{1(0,0)} + 2\, \# {\cal D}_{\frac{1}{2}(0,\frac{1}{2})} \, ,\) 
where \(n^{\rm T}\equiv 0\)  if the Sugawara condition is obeyed and \(n^{\rm T}\equiv 1\)  if it is not obeyed.

